Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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Laplace's equation

I am very confused. So I have Laplace's equation $\nabla^2\phi(x,y)=0$ and B.C.'s $\phi(x,0)=f(x); \,\,\,\,\,\, \phi(x,1)\equiv0$ where I have to solve it by Fourier transform. So I take the ...
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161 views

On a duality Fefferman-Stein's inequality

Let $M$ be the Hardy-Littlewood maximal operator. In the book "Weighted norm inequalities and Related Topics" by Rubio de Francia and J. Cuerva, page 150, theorem 2.1.2 states as the following: *For ...
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Solving this Fourier transform?

Is there any way to compute in closed form (in terms of known functions) the Fourier integral $$ \int_{-\infty}^{\infty} \frac{\cos(ux)}{(x^{2}+a^{2})^{s}} dx$$ where $u$ and $a$ are real positive ...
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256 views

Fourier transform properties

In my engineering class, I was told that $(FF(f))(x)=2\pi f(-x) $, where $F$ is the Fourier transform ----(1) and $F(f(x-a))(k)=\exp(-ika) X(k)$ where $X(k)=F(f(x))$ ----(2) implies ...
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156 views

Fourier transform inequalities on a probability distribution

I am reading a paper and the following came up: Given a probability density function, $\rho(x)$, such that for $\epsilon > 0$ $$ \int_{-\infty}^{\infty} |\rho(x)|^{1+\epsilon}dx < \infty ...
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Duality and the Fourier transform

Regarding Fourier transform, I read that the translation property and frequency-shift property are a duality. What does that mean and why is it true? Is there a physical implications? Thanks.
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Fourier Transform of complicated product: $(1+x)^2 e^{-x^2/2}$

UPDATE: The problem reduces to this for me: I ran into an issue with a change of variable integral: I have $\int\limits_{-\infty}^{\infty}e^{-\frac{1}{2}(x+\frac{iy}{2})^{2}}dx$. I do the change of ...
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146 views

About coefficients of Fourier integrals

If a $2\pi$-periodic function $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lebesgue integrable in $[-\pi,\pi]$, and the series $\frac{a_0}{2}+\sum_{n=1}^\infty [a_n \cos{nx}+b_n \sin{nx}] $, where ...
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115 views

inequality with roots of unity

Do you know proofs or references for the following inequality: There exists a positive constant $C>0$ such that for any complex numbers $a_1,\ldots,a_n$ $$ |a_1|+\cdots+|a_n| \leq ...
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329 views

Fourier transform of exp(exp(x))

I am interested in the Fourier transform of a function of the form $f(x) = \exp(g(\exp(x)))$, where g has a "simple" form, for example $g(y) = \frac{(y-1)^2}{y^2 - 1}$. Has anyone a starting point ...
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158 views

Fourier transform in $\mathbb R^3$

I try to show that $$ \int\limits_{R^3} \frac{e^{i\xi x} d\xi}{\xi^2 - k^2 - i0} = e^{ikx} \int\limits_{R^3} \frac{e^{i\xi x}d\xi}{\xi^2 + 2(k + i0\frac{k}{|k|})\xi}, \;\;\; k,x \in \mathbb R^3 $$ ...
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151 views

If a Fourier Transform is continuous in frequency, then what are the “harmonics”?

The basic idea of a Fourier series is that you use integer multiples of some fundamental frequency to represent any time domain signal. Ok, so if the Fourier Transform (Non periodic, continuous in ...
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46 views

Bessels and initial conditions

I'd like to know if I have got the following ideas right: 1) $f(r,\theta,t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty a_{nk}J_n(j_{nk}r)\exp[in\theta-j^2_{nk}t]$ subjected to initial ...
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83 views

Fourier analysis confusion

I think I may have misinterpreted this question, anyhow I am very confused. Here it is in its full glory: Let $f(r,\theta, t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty ...
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346 views

Fourier Transform of Schwartz Space

I am trying to read through Corollary 8.23 in Folland, p. 250, which is a proof that the Fourier transform maps the Schwartz space into itself. I do not see why the following is true $$\|x^\alpha ...
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Fourier transform of Bessel functions

I'm curious as to how the Fourier transform of the various types of Bessel functions would be calculated. The Wikipedia page on the Fourier transform gives the transform of $J_o(x)$ as being ...
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226 views

Conditions for the Convolution $f \ast g$ to be Continuous at a Point

Let $f$ and $g$ be functions on $\mathbb{R}^n$. Let $x_0$ be a given point in the unit ball $B(0,1)$. I am looking for sufficient conditions for the convolution $$ (f \ast g)(x) = \int_{B(0,1)} ...
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Generalized Fourier series

This is my first post here, so I hope I'm not doing sth wrong. I have to prove the following statement: For $f \in C^1(\mathbb{R^2})$ define $f_n(x,y) = \int_{\mathbb{T}} f(k(\theta)(x,y))e^{-2\pi i ...
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454 views

pointwise convergence of Fourier series

I am a bit confused. I have heard today someone saying that the Fourier series of any continues periodic function $f$, say with period 1 for concreteness, converges pointwise to $f$. Wikipedia here ...
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Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$

I'm trying to compute $$\int_{-\infty}^\infty \frac{\sinh(kx)}{\sinh(x)}e^{-i\omega x} \ dx$$ i.e. the Fourier transform of $x\mapsto \frac{\sinh(kx)}{\sinh(x)}$, where $0<k<1$ is fixed. But ...
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Non-power-of-2 FFT's?

If I have a program that can compute FFT's for sizes that are powers of 2, how can I use it to compute FFT's for other sizes? I have read that I can supposedly zero-pad the original points, but I'm ...
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285 views

Approximation with complex polynomials on $S^1$ - can it be done?

Can one uniformly approximate a function 'similar' to identity on $S^1$ with complex polynomials? I mean a function like: $f(z)=z \cdot (1+h \cdot \sin(m\cdot Arg(z)))$, for $|h| < 1,\ m \in ...
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For what sequences of real numbers $\left\{ k_{n}\right\}$ is the set of functions $\left\{ e^{ik_{n}x}\right\}$ a basis?

It is well known that the set of functions $\left\{ e^{^{inx}}\right\}$, for integer $n$, is an othonormal basis for the space of square integrable real functions in the interval $[-\pi,\pi]$. Now ...
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128 views

Find the probability of certain measurement for a Laplace Operator on a state function

Let $H$ be the operator $ -\frac{d^{2}}{dx^{2}} $ and let its domain be $$\{f\in L^{2}(\mathbb{R},d\lambda)\text{ }:\int_{-\infty}^{\infty}|xF[f(x)]|^{2}dx<\infty\} $$ where $F$ is the Fourier ...
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476 views

For symmetric stable distributions, why is $\alpha \le 2$?

I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact. Suppose we are trying to come up with stable distributions. From the definition, ...
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148 views

Product of Sidon sets

Let $G$ be a compact abelian group with dual $\Gamma$. Let $\Lambda \subset \Gamma$ a Sidon set (see the book of Rudin: Fourier Analysis on Groups for the definition). Consider the set ...
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756 views

Fourier transform inversion formula

We know that if $\hat f(\omega)=\frac{1}{2\pi}\int_{-\infty}^\infty f(t)e^{-i\omega t}dt$ is the Fourier transform of $f(t)$ then under some conditions, $f(t)=\int_{-\infty}^\infty \hat ...
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318 views

Heisenberg Uncertainty Principle

The uncertainty principle (UP) comes up in engineering and physics, but it is a mathematical idea. An old text describes it as "reciprocal spreading." If $f$ is a well-behaved function, the UP might ...
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199 views

Example of a sequence of uniformly but not absolutely convergent functions

(Background definitions) $A(\mathbb{T})$ is the space of all $2\pi$ periodic functions $f$ such that $\sum\limits_{k=-\infty}^{\infty}|\widehat{f}(k)| < \infty$. It is a normed space when normed ...
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Applications of Fourier analysis for math students

Does anyone have recommendations for a good book which explains how Fourier analysis is applied in engineering and physics that does not assume any knowledge of the above topics? I'd like to see the ...
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Why Pólya's method is wrong?

By truncating the Fourier transform, Pólya managed to prove that the Xi function on the critical line was approximately $$\xi(1/2+is) = (2\pi)^2 ( K_{9/4+is/2}( 2\pi) +K_{9/4-is/2}( 2\pi))$$ If this ...
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Condition for Fourier series

I read that Any "well-behaved" function of period $2\pi$ can be expressed as a Fourier series. What qualifies as "well-behaved"? Any examples of functions that cannot be expressed as a ...
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Fourier transform of unit step?

I don't understand what's wrong with my derivation below... $\delta(t) = u'(t)$ $\mathcal{F}(\delta)(\omega) = 1 = \mathcal{F}(u')(\omega) = i\omega \times \mathcal{F}(u)(\omega)$ (since the ...
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1answer
277 views

Representation of Fourier Transform's vectors

I am just learning FFTs and I am trying to debug a problem in MATLAB. I think I don't understand how is MATLAB's FFT function handling the polynomial powers, or I am doing something wrong manually. ...
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111 views

Are these Fourier transforms equal?

I believe that since $|x|^2=x^2$ then we have the Fourier transforms $$\int_{-\infty}^{\infty} \mathrm dx \frac{\exp{iux}}{a^2+|x|^2} =\int_{-\infty}^{\infty}\mathrm dx \frac{\exp{iux}}{a^2+x^2}$$ ...
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Inequality for Trigonometric Polynomials

Problem statement: Define $p(t) = \sum\limits_{j=-N}^{N}c_{j}e^{ijt}$ be a real-valued trigonometric polynomial. Suppose there exists an $x_{0}\in\mathbb{R}$ such that $p(x_{0}) = \|p\|_{\infty}$. ...
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258 views

$L^1$ norm of the Fourier transform of a truncated Gaussian

Consider the Gaussian $G(x):=e^{-x^2}$ on the real line, and localize it to the region $|x|\sim 2^k$ by multiplying it by an appropriate smooth cut-off. More precisely, take $\phi\in ...
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327 views

Analytic continuation of Fourier transform

Let $$h(u)= \int_{-\infty}^{\infty} g(x) \exp(iux) \, \text{d}x$$ be the Fourier transform. Then let us suppose that for $ |x| \to \infty $ the function $g$ goes as $$g(x) = \exp(-ax) \text{ for ...
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208 views

Fourier transform

Suppose $1< p<\infty$. Let $f$ be a continuous function with compact support defined on $\mathbb{R}$. Does it exist a function $g \in L^p(\mathbb{T})$ such that: $$ ...
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117 views

Asymptotics of an improper integral

I have to show that if $x \to \infty$, then $$ \int\limits_{\mathbb{R}^d} \frac{e^{i\xi x}}{\xi^2 + 2k\xi}d\xi = O\left(|x|^{-\frac{d-1}{2}} \right) \;\;\; \; d\geqslant2, \;\;\; k\in \mathbb{C}^d ...
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704 views

Finding the support of the convolution of two functions

This is a rather long problem and I'm having difficulties with a very specific part. The question starts with a function $\phi\in L_{1}(\mathbb{R})$ which vanishes outside of $(-\pi,\pi)$. Then for ...
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335 views

Proving a function is Lipschitz

I have a homework problem which consists of two parts, the first of which I have been staring at for several days with very little (constructive) progress. I need to show that the function $$f(t) = ...
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An estimate on the $p$ norm of a convolution integral

The theorem is that for any summability kernel $\{\phi_{n}\}$, if $f\in L_{p}(\mathbb{T}^{d})$, then $||f*\phi_{n} - f||_{p}\rightarrow 0$. The step that I cannot follow is this: ...
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Convolution of an $L_{p}(\mathbb{T})$ function $f$ with a term of a summability kernel $\{\phi_n\}$

... is the result in $L_{p}$? A remark in my notes says yes but I can't see how to verify it. As was pointed out to me in a previous question I asked last night, I need to show that the following ...
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102 views

Is $L_{p}(\mathbb{T})\subset L_{\infty}(\mathbb{T})$?

I was about to post yet another question about a comment in my notes. But I think my key point in misunderstanding jumps in proofs is how the $L_{p}$ spaces are nested in eachother. The example that ...
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230 views

$L_{p}$ distance between a function and its translation

I'm working through a proof and one of the comments is that for a function $f\in L_p (\mathbb{T})$: $$\lim_{t\to \infty}\;\|f(\cdot + t) - f\|_p = 0.$$ Should this read as $t\to 0$? If so, how do ...
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301 views

Convolution Integral only defined Almost Everywhere?

My professor has mentioned in class that if we have two $2\pi$-periodic functions $f$ and $g$ that are both in $L_1(\mathbb{T})$, then $$(f*g)(t) := \frac{1}{2\pi}\int_{-\pi}^{\pi}f(s-t)g(s)ds$$ is ...
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210 views

Pointwise convergence of Fourier Series of functions of bounded variation

Another question from a theorem in my notes: Let $f\in BV(\mathbb{T})$. Then for every $x\in\mathbb{R}$, $S_{n}(f)(x)\to \dfrac{f(x+0) + f(x-0)}{2}$ (and to $f(x)$ at every point $x$ where $f$ is ...
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126 views

Pointwise convergence of the Cesàro mean of a function

A theorem in my notes claims the following: If $f\in L_1(\mathbb{T})$, and both $f(x+0) = \lim\limits_{t\to x^+} f(t)$ and $f(x - 0) = \lim\limits_{t\to x^-} f(t)$ exist, then ...
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707 views

Why is the Fejér Kernel always non-negative?

In one of the proofs in my notes the Fejér Kernel ($K_{n}$ below) is plucked out of an absolute value seemingly for free. On the previous page it is remarked that this function is always ...